Circular segment

In geometry, a circular segment (symbol: ⌓) is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc (of less than 180°) of a circle and by the chord connecting the endpoints of the arc.

Formulae

A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area).

Let R be the radius of the circle, θ is the central angle in radians, α is the central angle in degrees, c the chord length, s the arc length, h the height of the segment, and d the height of the triangular portion.

• The radius is $R=h+d=h/2+c^{2}/8h{\frac {}{}}$
• The arc length is $s={\frac {\alpha }{180}}\pi R={\theta }R$

• The chord length is $c=2R\sin {\frac {\theta }{2}}=R{\sqrt {2-2\cos \theta }}$

• The height is $h=R\left(1-\cos {\frac {\theta }{2}}\right)=R-{\sqrt {R^{2}-{\frac {c^{2}}{4}}}}$

• The angle is $\theta =2\arctan {\frac {c}{2d}}=2\arccos {\frac {d}{R}}=2\arcsin {\frac {c}{2R}}$

Area

The area of the circular segment is equal to the area of the circular sector minus the area of the triangular portion—that is,

$A={\frac {R^{2}}{2}}\left(\theta -\sin \theta \right).$

or with the central in degrees,

$A={\frac {R^{2}}{2}}\left({\frac {\alpha \pi }{180}}-\sin \alpha \right).$

See also

• Spherical segment
• Conic section
• Cross section

References

• Weisstein, Eric W., "Circular segment", MathWorld.

External links

• Definition of a circular segment With interactive animation
• Formulae for area of a circular segment With interactive animation